Lecture Notes CS:5350 Introduction to Greedy approximation Algorithms Lecture 6: Nov. 7, 2019 Scribe: Qi Qi

1 Minimum Cover

Greedy algorithm for MVC (Here is a greedy (deterministic) algorithm for MVC): 1. S ← ∅ 2. while (S is not a vertex cover) do: // greedy step 3. pick a vertex with highest degree, v, in active graph and add to S 4. deactivate all activate edges incident on v 5. Output S

1.1 Counter Examples: Figure 1 shows a simple counter example to show that the greedy algorithm will not always produce an optimal solution for MVC.

Figure 1: Simple counterexamp graph

In fact, the output of the greedy algorithm can be quite poor, as showing in the following claim. Claim: There exists a graph G = (V,E), |V | = n such that |S| = Ω(log(n)) · |S∗|, where S is the output of greedy algorithms, and S∗ is the optimal vertex cover. Proof: Construct a (Figure 2), which contains G = (L ∪ R,E). Let k denote |L|, where R = R1 ∪ R2 ∪ · · · Rk. For each set Ri: k 1. |Ri| = b i c

2. Each vertex in Ri has degree i and no two vertexes in Ri have a common neighbour.

The vertex in Rk would be the first node to to be selected to join S by the greedy MVC algorithm since its degree is k. All other nodes in R have degree less than k and nodes in L also have degree less than k (the largest degree in the L is k − 1). After deleting all the edges that connected with this node Rk, all the degree of nodes in the L will also decrease 1. Then in the next iterations, the nodes in Rk−1 will be selected one by one, and on and on until all the nodes in R are selected to join S. Pk k Pk 1 Then the final output |S| = i=2 |Ri| = b i c| ∼ k i=1 i = Θ(k log(k)), but |L| = k. .

1 Figure 2: Constructed counterexample of bipartite graph

2 Landscape of problems approximation factors

Category of Best known Approximation Factor Approximation Factor Problem Notes Using data rounding and dynamic programming, there PTAS (1 + ), for any  > 0 Knapsack n3 is an O(  ) algorithm α-approximation, 2 K-center for α < 2 is not possible unless P = NP constant Whether there is a better than 2 MVC 2-approximation is a long- standing open problem Better approximation is not ln(n) SET COVER possible unless all problems in NP n =size of ground set can be solved in sub-exponential time logarithmic ln(n) Min. Dominating MDS is just a special n =size of vertexes Set (MDS) case of SET COVER 1− Maximum No O(n )-approximation exists n Polynomial O( poly(log n) ) Independent Set unless P=NP Table 1: Landscape of problems approximation factor

Figure 3: Landscape of techniques

Remark: Figure 3 illustrates the landscapes of techniques for solving the problems in Table 1.

2 3 K-Center

K-Center: well known clustering problems with many applications (e.g. unsupervised learning.)

Definition (Distance Metric): Let P = {p1, p2, ··· , pn} be a set of points. Let D : P ×P → R≥0 be a function. D is a called metric if:

1. D(pi, pi) = 0 for all pi ∈ P (reflexive)

2. D(pi, pj) = D(pj, pi) for all pi, p = j ∈ P (symmetric)

3. D(pi, pj) + D(pj, pk) ≥ D(pi, pk) for all pi, pj, pk ∈ P (triangle inequality) Examples (Aside from Euclidean Distance): 1. Let G = (V,E) be an undirected graph. for any u, v ∈ V , D(u, v) denotes the shortest path distance between (u, v). It is easy to check that D is a metric. 2. Let G = (V,E) be an undirected graph. For any u, v ∈ V , let D(u, v) = 1, if {u, v} ∈ E and D(u, v) = 2 if {u, v} ∈/ E. Also D(v, v) = 0 for all v ∈ V . D is a metric as D(u, v)+D(v, k) ≥ D(u, k).

Notation: Let S ⊆ P , D(p, S) = mins∈S D(p, s):

Figure 4: Illustrates of the distance between a point to a set i.e., D(p, S).

K-Center:

1. Input: A metric D : P × P → R≥0.

2. Output: A subset S ⊆ P, |S| = k such that maxp∈P D(p, S) is minimized. Alternative view of K-Center: Let S ⊆ P , |S| = k. For each p ∈ P , assign p to the nearest ¨center¨, (i.e point in S). And for each 0 s ∈ S, Ball(s) = set of points assigned to s. And the radius of s is radius(s) = maxp0∈Ball(s) D(p , s). And the radius of the set S Radius(S) = maxs∈S radius(s). We are looking for a subset S ⊆ P with k points, whose radius is the smallest.

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